The ordered elements in several one-dimensional Coulomb gas ensembles arising in probability and mathematical physics are shown to have log-concave distributions. Examples include the beta ensembles with convex potentials (in the continuous setting) and the orthogonal polynomial ensembles (in the discrete setting). In particular, we prove the log-concavity of the Tracy-Widom β distributions, Airy distribution, and Airy-2 process. Log-concavity of last passage times in percolation is proven using their connection to Meixner ensembles. We then obtain the log-concavity of top rows of Young diagrams under Poissonized Plancherel measure, which is the Poissonized version of a conjecture of Chen. This is ongoing joint work with Jnaneshwar Baslingker and Manjunath Krishnapur.